1066 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 5, SEPTEMBER 2008 Enhancing Controller Performance for Robot Positioning in a Constrained Environment Franco Blanchini, Stefano Miani, Felice Andrea Pellegrino, and Bart van Arkel Abstract—The paper deals with the problem of positioning a manipulator in a cluttered environment while avoiding collision with obstacles. Recently a control strategy based on invariant sets has been introduced by some of the authors: it consists of covering the configuration space by means of a connected family of poly- hedral regions which can be rendered controlled-invariant. Each of these regions includes some crossing points to the confining (and partially overlapping) regions. The control is hierarchically structured: a high-level controller establishes a proper sequence of regions to be crossed to reach the one in which the target con- figuration is included. A low-level controller solves the problem of tracking, within a region, the crossing point to the next con- fining region and, eventually, tracking the reference whenever it is included in the current one. Here we focus on the low-level controller, providing two novel contributions: first we extend the previous results, based on a vertex representation of the polyhedral sets, to the face representation which is more natural and offers significant computational advantages for on-line implementation; second, we provide a new low-level speed-saturated controller in order to improve the performance of the previous one in terms of convergence speed. We also investigate the robustness of the proposed controller. Experimental results on a Cartesian robot are provided. Index Terms—Hierarchical systems, invariance, Lyapunov methods, manipulator motion-planning, robustness. I. INTRODUCTION N AVIGATING in a cluttered environment while avoiding obstacles is important for robots in general and for ma- nipulators in particular. Several techniques have been proposed in the past to deal with this problem [7], [14], [11], [16], [1], [20], [13], [8], [10]. Some of them focus on the pure geometrical problem of finding an admissible path, discarding the problem of tracking that path in a dynamically feasible manner. Some others bring the control into the picture, trying to find a control law that results in a feasible path. A new technique [5], falling in the latter category, was proposed recently by some of the au- thors. It consists of tracking a suitable sequence of reference signals in an admissible set described by the union of partially overlapping polyhedra. In the mentioned paper the (constrained) positioning problem was solved by means of a two-level control scheme (an approach pursued also in [6], [15]), with the high Manuscript received November 27, 2006; revised August 27, 2007. Manu- script received in final form November 14, 2007. First published March 31, 2008; last published July 30, 2008 (projected). Recommended by Associate Ed- itor M. de Mathelin. F. Blanchini is with the Department of Mathematics and Computer Science, University of Udine, 208-33100 Udine, Italy (e-mail: blanchini@uniud.it). F. A. Pellegrino is with the Department of Electrical, Electronic and Com- puter Engineering, University of Trieste, 34100 Trieste, Italy (e-mail: fapelle- grino@units.it). S. Miani and B. van Arkel are with the Department of Electrical, Managerial and Mechanical Engineering, University of Udine, 33100 Udine, Italy (e-mail: miani.stefano@uniud.it; vanarkel@uniud.it). Digital Object Identifier 10.1109/TCST.2007.916324 level taking care of generating an appropriate sequence of ref- erence signals and the low-level controller tracking the current reference while guaranteeing constraints satisfaction. Here we reconsider this technique and present new results concerning the computation of the low-level controller and its application to manipulators. In particular, in Section III we show that the low-level controller can be obtained by resorting to the face representation of the polyhedra instead of the vertex repre- sentation, as done in the previous work. This leads to a signifi- cant advantage in terms of (especially on-line) computation. We reconsider the control scheme and we derive the dual version of the set-based controller previously proposed. In Section IV we point out a severe drawback of the approach, in terms of per- formances, due to the linear nature of the low-level controller. Therefore, we propose a new saturated low-level control law which assures a better behavior. The implementation of this con- trol is very simple in the speed-control case. Conversely, in the torque control case, it turns out to be more involved and re- quires a proper backstepping procedure. Finally, in Section V we present some experimental results on a Cartesian robot. II. PROBLEM FORMULATION Consider the fully actuated mechanical manipulator de- scribed by the dynamic system (1) where is the vector of Lagrangian coordinates, is the input vector, represents the Coriolis and centrifugal terms, represents the gravitational terms, and is the inertia matrix, which is assumed invertible for every . Following [18] we call the space of configurations C-space. Due to the presence of obstacles and to the structure of the manipulator itself, only a subset of the C-space, the free C-space, is allowed for the manipulator to move in. The aim is that of reaching a target belonging to the free C-space while avoiding collisions. Hence, the controller has to guarantee that the whole trajectory remains in the free C-space. 1 Consider the subset of the free C-space (2) where the sets are polyhedra. We assume that the family of the interiors of these sets forms a connected family so that each two points in the interior of can be connected by a continuous curve included in the union of the interiors of some sequence of . The basic problem considered here is the following: 1 Actually, the results presented here hold for a more general problem, namely that of tracking a time-varying reference such that . 1063-6536/$25.00 © 2008 IEEE